User:IssaRice/Linear algebra/Rank of polynomial matrix is constant everywhere except possibly at finitely many points: Difference between revisions
(Created page with "This is Corollary 6.2 in ''Linear Algebra Done Wrong''. I find the proof in the book pretty unclear, so I want to write up a clearer proof. Corollary statement: Let <math>A(...") |
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I find the proof in the book pretty unclear, so I want to write up a clearer proof. | I find the proof in the book pretty unclear, so I want to write up a clearer proof. | ||
Corollary statement: Let <math>A(x)</math> be an <math>m \times n</math> polynomial matrix (i.e. a matrix whose entries are polynomials of <math>x</math>). Then <math>\operatorname{rank} A(x)</math> is constant everywhere, except possibly at finitely many points, where the rank is smaller. | Corollary statement: Let <math>A(x)</math> be an <math>m \times n</math> polynomial matrix (i.e. a matrix whose entries are polynomials of <math>x</math>). Then <math>x \mapsto \operatorname{rank} A(x)</math> is constant everywhere, except possibly at finitely many points, where the rank is smaller. | ||
Proof: | Proof: | ||
Revision as of 04:14, 16 December 2020
This is Corollary 6.2 in Linear Algebra Done Wrong.
I find the proof in the book pretty unclear, so I want to write up a clearer proof.
Corollary statement: Let be an polynomial matrix (i.e. a matrix whose entries are polynomials of ). Then is constant everywhere, except possibly at finitely many points, where the rank is smaller.
Proof: